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Structure-Preserving Flows of Symplectic Matrix Pairs

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Structure-Preserving Flows of Symplectic Matrix Pairs STRUCTURE-PRESERVING FLOWS OF SYMPLECTIC MATRIX PAIRS YUEH-CHENG KUO∗, WEN-WEI LINy , AND SHIH-FENG SHIEHz Abstract. We construct a nonlinear differential equation of matrix pairs (M(t); L(t)) that are invariant (Structure-Preserving Property) in the class of symplectic matrix pairs X12 0 IX11 (M; L) = S2; S1 X = [Xij ]1≤i;j≤2 is Hermitian ; X22 I 0 X21 where S1 and S2 are two fixed symplectic matrices. Furthermore, its solution also preserves de- flating subspaces on the whole orbit (Eigenvector-Preserving Property). Such a flow is called a structure-preserving flow and is governed by a Riccati differential equation (RDE) of the form > > W_ (t) = [−W (t);I]H [I;W (t) ] , W (0) = W0, for some suitable Hamiltonian matrix H . We then utilize the Grassmann manifolds to extend the domain of the structure-preserving flow to the whole R except some isolated points. On the other hand, the Structure-Preserving Doubling Algorithm (SDA) is an efficient numerical method for solving algebraic Riccati equations and nonlinear matrix equations. In conjunction with the structure-preserving flow, we consider two special classes of sym- plectic pairs: S1 = S2 = I2n and S1 = J , S2 = −I2n as well as the associated algorithms SDA-1 k−1 and SDA-2. It is shown that at t = 2 ; k 2 Z this flow passes through the iterates generated by SDA-1 and SDA-2, respectively. Therefore, the SDA and its corresponding structure-preserving flow have identical asymptotic behaviors. Taking advantage of the special structure and properties of the Hamiltonian matrix, we apply a symplectically similarity transformation to reduce H to a Hamiltonian Jordan canonical form J. The asymptotic analysis of the structure-preserving flows and RDEs is studied by using eJt. Some asymptotic dynamics of the SDA are investigated, including the linear and quadratic convergence. 1. Introduction. The following three important types of algebraic Riccati equa- tions appear in many different fields of applied mathematics, sciences and engineering. • Continuous-time Algebraic Riccati Equation (CARE) [24, 28]: −XGX + AH X + XA + H = 0; (1.1) • Discrete-time Algebraic Riccati Equation (DARE) [24, 28]: X = AH X(I + GX)−1A + H; (1.2) • Nonlinear Matrix Equation (NME) [11]: X + AH X−1A = Q; (1.3) where A; H; G; Q 2 Cn×n with G = GH , H = HH and Q = QH . The CARE (1.1) and DARE (1.2) with G and H being positive semi-definite arise in the linear quadratic (LQ) optimal control problems in continuous- and discrete-time, respectively (see [24, 28]). In the H1-control problems, G and H in (1.1) or (1.2) can possibly be indefinite [13, 28]. In addition, solutions of a CARE are the equilibria of Riccati differential equations (RDEs) that arise frequently from optimal controls [23, 25], or two-point boundary value problems [9, 10]. The NME (1.3) occurs in applications such as analysis of ladder networks, dynamic programming and stochastic filtering ∗Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 811, Taiwan ([email protected]) yDepartment of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan ([email protected]) zDepartment of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan ([email protected]) 1 (see [2, 11, 12] and the references therein). Nonclassical Riccati-type equations for which the conjugate transpose in (1.3) is replaced by the complex transpose also have various applications; for example, in the vibration analysis of fast trains [16] and the Green's function calculation in nano research [15]. The Structure-Preserving Doubling Algorithms (SDAs) [3, 20, 27] are employed for solving the stabilizing solutions of the three Riccati-type equations (1.1)-(1.3). We now describe these SDAs for solving DARE/CARE and NME. Ak 0 IGk • For solving DAREs (1.2), the pairs (Mk; Lk) = ; H −Hk I 0 Ak are generated by Algorithm SDA-1 [20, 27]. Let A1 = A, G1 = G, and H1 = H. For k = 1; 2;:::, compute −1 Ak+1 = Ak(I + GkHk) Ak; −1 H Gk+1 = Gk + AkGk(I + HkGk) Ak ; H −1 Hk+1 = Hk + Ak (I + HkGk) HkAk: • For solving CAREs (1.1), one can transform it into a DARE (1.2) by using the Cayley transform and a left equivalence transform [4]. Then Algorithm SDA-1 can be employed to find the desired solutions of CAREs. Ak 0 −Pk I • For solving NMEs, the pairs (Mk; Lk) = ; H are Qk −I Ak 0 generated by Algorithm SDA-2 [3, 27]. Let A1 = A, Q1 = Q, and P1 = 0. For k = 1; 2;:::, compute −1 Ak+1 = Ak(Qk − Pk) Ak; H −1 Qk+1 = Qk − Ak (Qk − Pk) Ak; −1 H Pk+1 = Pk + Ak(Qk − Pk) Ak : 1 Assume that the sequences f(Mk; Lk)gk=1 generated by the SDAs are well- 1 1 defined. The sequences f(Hk;Gk;Ak)gk=1 by SDA-1 and f(Qk;Pk;Ak)gk=1 by SDA-2 ∗ ∗ ∗ ∗ converge quadratically to (H ;G ; 0) and (Q ;P ; 0), respectively, if (M1; L1) has no unimodular eigenvalues [3, 27]. The sequences by SDA-1 and SDA-2 converge linearly of the rate 1=2 if the partial multiplicities of the unimodular eigenvalues of (M1; L1) are all even [3, 20]. Here H∗ and G∗ are the stabilizing solutions of (1.2) and its dual equation: Y = AY (I + HY )−1AH + G, respectively, with the spectral radius ρ((I + GH∗)−1A) < 1. Q∗ and P ∗ are the maximal and the minimal solutions of (1.3), respectively, with Q∗ − X and X − P ∗ being positive semi-definite, where X is any solution of (1.3). Furthermore, the SDA has two preserving properties [22, 27]: Eigenvector-Preserving Property: For each case above, if MkU = LkUS or 2n×r r×r MkVT = LkV , where U; V 2 C are of full column rank and S; T 2 C , 2 2 then Mk+1U = Lk+1US or Mk+1VT = Lk+1V , i.e., the SDA preserves the deflating subspaces for each k and squares the eigenvalue matrix; 1 Structure-Preserving Property: The sequences of pairs f(Mk; Lk)gk=1 gener- ated by SDA-1 and SDA-2 are symplectic (see Definition 2.1) and are, re- spectively, contained in the sets A 0 IG = ; jA; H = HH ;G = GH 2 n×n ; (1.4a) S1 HI 0 AH C 2 and A 0 PI = ; jA; Q = QH ;P = P H 2 n×n : (1.4b) S2 Q −I AH 0 C To study the symplectic pairs, we first quote the following useful theorem. Theorem 1.1 (see [29]). Let (M; L) be a regular symplectic pair with M; L 2 2n×2n C . Then there exist symplectic matrices S1, S2 and a Hermitian matrix X = X11 X12 l:e: X12 0 IX11 such that (M; L) ∼ S2; S1 . Here we say X21 X22 X22 I 0 X21 l.e. (A1;B1) ∼ (A2;B2), if A1 = CA2, B1 = CB2 for some invertible matrix C. Theorem 1.1 provides us a classification for symplectic pairs. Specifically, let S1; S2 be symplectic and H(2n) be the set of all 2n × 2n Hermitian matrices. We denote the class of symplectic pairs generated by S1; S2 as X12 0 IX11 X11 X12 SS1;S2 = S2; S1 j 2 H(2n) : (1.5a) X22 I 0 X21 X21 X22 It is easily seen that each pair (M; L) 2 SS1;S2 is symplectic. We define a bijective transformation TS1;S2 : H(2n) ! SS1;S2 by X12 0 IX11 TS1;S2 (X) = S2; S1 : (1.5b) X22 I 0 X21 Therefore, the invariant sets S1 and S2 for SDA-1 and SDA-2, respectively, can be rewritten as S1 = SI2n;I2n and S2 = SJ ;−I2n . Note that S1 *S2, S2 *S1. The purpose of this paper is to focus on the flows (M(t); L(t)) on a specified SS1;S2 (i.e., the Structure-Preserving Property) that have Eigenvector-Preserving Property. Such a flow is called a structure-preserving flow. For two special classes S1 and S2 of symplectic pairs, the structure-preserving flow passes through a sequence of iterates generated by SDAs. Finding a smooth curve with a specific structure that passes through a sequence of iterates generated by some numerical algorithm is a popular topic studied by many researchers, especially in the study of the so- called Toda flow that links matrices/matrix pairs generated by QR/QZ-algorithm [5, 6, 7, 32]. In [22], a parameterized curve is constructed in S2 passing through the iterates generated by the fixed-point iteration, the SDA and the Newton's method with some additional conditions. Our first main contribution in this paper is to construct a nonlinear differen- tial equation associated with the structure-preserving flow satisfying (M(1); L(1)) = (M1; L1) 2 SS1;S2 . Before the description of this main result, we first quote the pre- liminary result in Theorem 2.7. Suppose that ind1(M1; L1) 6 1. There exist an 2n×2^n 2n×` 2n×` invertible matrix U = [U1jU0;U1] 2 C × C × C and a Hamiltonian 2^n×2^n Hb Hb 2 C such that M1U0 = 0, L1U1 = 0, and M1U1 = L1U1e . Here, the spaces spanned by U0, U1, and U1 represent the eigenspaces corresponding to zero eigenvalues, infinity eigenvalues, and finite nonzero eigenvalues of the pair, respec- tively. Main Result I (Theorem 3.3). Let (M1; L1) 2 SS1;S2 be a regular symplectic pair with ind (M ; L ) 1 and X = T −1 (M ; L ). There exists a Hamiltonian 1 1 1 6 1 S1;S2 1 1 H 2 C2n×2n such that the solution of the initial value problem (IVP): H X_ (t) = M(t)HJ M(t) ;X(1) = X1; (1.6) 3 Hbt for t 2 (t0; t1), satisfies M(t)U0 = 0, L(t)U1 = 0 and M(t)U1 = L(t)U1e , where (M(t); L(t)) = TS1;S2 (X(t)).
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